At one end A of a diameter `A B` of a circle of radius 5cm, tangent `X A Y` is drawn to the circle. The length of chord `C D` parallel to `X Y` and at a distance of 8cm from `A` is

Options

A. 5cm
B. 6cm
C. 7cm
D. 8cm

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the Problem
We have a circle with a radius of 5 cm and a diameter AB. A tangent line XY is drawn at point A, and we need to find the length of the chord CD that is parallel to the tangent XY and located 8 cm away from point A.

### Step 2: Draw the Diagram
1. Draw a circle with center O and radius 5 cm.
2. Mark the diameter AB, where A is one end of the diameter.
3. Draw the tangent line XY at point A.
4. Mark point N on line OA such that AN = 8 cm, which is the distance from A to the chord CD.

### Step 3: Determine the Distance from the Center to the Chord
1. Since the radius OA = 5 cm, we can find the distance ON from the center O to the line AN.
2. We know AN = 8 cm and AO = 5 cm.
3. Therefore, ON = AN - AO = 8 cm - 5 cm = 3 cm.

### Step 4: Apply the Perpendicularity Rule
1. According to the property of circles, the radius is perpendicular to the tangent at the point of contact.
2. Therefore, OA is perpendicular to the chord CD, which means ON is also perpendicular to CD.

### Step 5: Use the Pythagorean Theorem
1. In the right triangle OCN, where OC is the radius (5 cm), ON is the distance from the center to the chord (3 cm), and CN is half the length of the chord CD.
2. By the Pythagorean theorem, we have:
\[ OC^2 = ON^2 + CN^2 \]
Substituting the known values:
\[ 5^2 = 3^2 + CN^2 \]
\[ 25 = 9 + CN^2 \]
\[ CN^2 = 25 - 9 = 16 \]
\[ CN = 4 \text{ cm} \]

### Step 6: Find the Length of Chord CD
1. Since N is the midpoint of chord CD, the full length of the chord CD is:
\[ CD = 2 \times CN = 2 \times 4 = 8 \text{ cm} \]

### Final Answer
The length of chord CD is **8 cm**.

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